Dimensionless fluctuations balance applied to statistics and quantum physics.

This work presents a new method called Dimensionless Fluctuation Balance (DFB), which makes it possible to obtain distributions as solutions of Partial Differential Equations (PDEs). In the first case study, DFB was applied to obtain the Boltzmann PDE, whose solution is a distribution for Boltzmann gas. Following, the Planck photon gas in the Radiation Law, Fermi-Dirac, and Bose-Einstein distributions were also verified as solutions to the Boltzmann PDE. The first case study demonstrates the importance of the Boltzmann PDE and the DFB method, both introduced in this paper. In the second case study, DFB is applied to thermal and entropy energies, naturally resulting in a PDE of Boltzmann's entropy law. Finally, in the third case study, quantum effects were considered. So, when applying DFB with Heisenberg uncertainty relations, a Schrödinger case PDE for free particles and its solution were obtained. This allows for the determination of operators linked to Hamiltonian formalism, which is one way to obtain the Schrödinger equation. These results suggest a wide range of applications for this methodology, including Statistical Physics, Schrödinger's Quantum Mechanics, Thin Films, New Materials Modeling, and Theoretical Physics.

Non-Additive Entropic Forms and Evolution Equations for Continuous and Discrete Probabilities.

Increasing interest has been shown in the subject of non-additive entropic forms during recent years, which has essentially been due to their potential applications in the area of complex systems. Based on the fact that a given entropic form should depend only on a set of probabilities, its time evolution is directly related to the evolution of these probabilities. In the present work, we discuss some basic aspects related to non-additive entropies considering their time evolution in the cases of continuous and discrete probabilities, for which nonlinear forms of Fokker-Planck and master equations are considered, respectively. For continuous probabilities, we discuss an H-theorem, which is proven by connecting functionals that appear in a nonlinear Fokker-Planck equation with a general entropic form. This theorem ensures that the stationary-state solution of the Fokker-Planck equation coincides with the equilibrium solution that emerges from the extremization of the entropic form. At equilibrium, we show that a Carnot cycle holds for a general entropic form under standard thermodynamic requirements. In the case of discrete probabilities, we also prove an H-theorem considering the time evolution of probabilities described by a master equation. The stationary-state solution that comes from the master equation is shown to coincide with the equilibrium solution that emerges from the extremization of the entropic form. For this case, we also discuss how the third law of thermodynamics applies to equilibrium non-additive entropic forms in general. The physical consequences related to the fact that the equilibrium-state distributions, which are obtained from the corresponding evolution equations (for both continuous and discrete probabilities), coincide with those obtained from the extremization of the entropic form, the restrictions for the validity of a Carnot cycle, and an appropriate formulation of the third law of thermodynamics for general entropic forms are discussed.

Nonlinear Fokker-Planck Equation Approach to Systems of Interacting Particles: Thermostatistical Features Related to the Range of the Interactions.

Nonlinear Fokker-Planck equations (NLFPEs) constitute useful effective descriptions of some interacting many-body systems. Important instances of these nonlinear evolution equations are closely related to the thermostatistics based on the S q power-law entropic functionals. Most applications of the connection between the NLFPE and the S q entropies have focused on systems interacting through short-range forces. In the present contribution we re-visit the NLFPE approach to interacting systems in order to clarify the role played by the range of the interactions, and to explore the possibility of developing similar treatments for systems with long-range interactions, such as those corresponding to Newtonian gravitation. In particular, we consider a system of particles interacting via forces following the inverse square law and performing overdamped motion, that is described by a density obeying an integro-differential evolution equation that admits exact time-dependent solutions of the q-Gaussian form. These q-Gaussian solutions, which constitute a signature of S q -thermostatistics, evolve in a similar but not identical way to the solutions of an appropriate nonlinear, power-law Fokker-Planck equation.

Equilibrium States in Two-Temperature Systems.

Systems characterized by more than one temperature usually appear in nonequilibrium statistical mechanics. In some cases, e.g., glasses, there is a temperature at which fast variables become thermalized, and another case associated with modes that evolve towards an equilibrium state in a very slow way. Recently, it was shown that a system of vortices interacting repulsively, considered as an appropriate model for type-II superconductors, presents an equilibrium state characterized by two temperatures. The main novelty concerns the fact that apart from the usual temperature T, related to fluctuations in particle velocities, an additional temperature θ was introduced, associated with fluctuations in particle positions. Since they present physically distinct characteristics, the system may reach an equilibrium state, characterized by finite and different values of these temperatures. In the application of type-II superconductors, it was shown that θ ≫ T , so that thermal effects could be neglected, leading to a consistent thermodynamic framework based solely on the temperature θ . In the present work, a more general situation, concerning a system characterized by two distinct temperatures θ 1 and θ 2 , which may be of the same order of magnitude, is discussed. These temperatures appear as coefficients of different diffusion contributions of a nonlinear Fokker-Planck equation. An H-theorem is proven, relating such a Fokker-Planck equation to a sum of two entropic forms, each of them associated with a given diffusion term; as a consequence, the corresponding stationary state may be considered as an equilibrium state, characterized by two temperatures. One of the conditions for such a state to occur is that the different temperature parameters, θ 1 and θ 2 , should be thermodynamically conjugated to distinct entropic forms, S 1 and S 2 , respectively. A functional Λ [ P ] ≡ Λ ( S 1 [ P ] , S 2 [ P ] ) is introduced, which presents properties characteristic of an entropic form; moreover, a thermodynamically conjugated temperature parameter γ ( θ 1 , θ 2 ) can be consistently defined, so that an alternative physical description is proposed in terms of these pairs of variables. The physical consequences, and particularly, the fact that the equilibrium-state distribution, obtained from the Fokker-Planck equation, should coincide with the one from entropy extremization, are discussed.

The Influence of Divalent Anions on the Rectification Properties of Nanofluidic Diodes: Insights from Experiments and Theoretical Simulations.

During the last decade, the possibility of generating synthetic nanoarchitectures with functionalities comparable to biological entities has sparked the interest of the scientific community related to diverse research fields. In this context, gaining fundamental understanding of the central features that determine the rectifying characteristics of the conical nanopores is of mandatory importance. In this work, we analyze the influence of mono- and divalent salts in the ionic current transported by asymmetric nanopores and focus on the delicate interplay between ion exclusion and charge screening effects that govern the functional response of the nanofluidic device. Experiments were performed using KCl and K2 SO4 as representative species of singly and doubly charged species. Results showed that higher currents and rectification efficiencies are achieved by doubly charged salts. In order to understand the physicochemical processes underlying these effects simulations using the Poisson-Nernst-Planck formalism were performed. We consider that our theoretical and experimental account of the effect of divalent anions in the functional response of nanofluidic diodes provides further insights into the critical role of electrostatic interactions (ion exclusion versus charge screening effects) in presetting the ionic selectivity to anions as well as the observed rectification properties of these chemical nanodevices.

Estimating vertical and lateral pressures in periodically structured montmorillonite clay particles

Given a montmorillonitic clay soil at high porosity and saturated by monovalent counterions, we investigate the particle level responses of the clay to different external loadings. As analytical solutions are not possible for complex arrangements of particles, we employ computational micromechanical models (based on the solution of the Poisson-Nernst-Planck equations) using the finite element method, to estimate counterion and electrical potential distributions for particles at various angles and distances from one another. We then calculate the disjoining pressures using the Van't Hoff relation and Maxwell stress tensor. As the distance between the clay particles decreases and double-layers overlap, the concentration of counterions in the micropores among clay particles increases. This increase lowers the chemical potential of the pore fluid and creates a chemical potential gradient in the solvent that generates the socalled 'disjoining' or 'osmotic' pressure. Because of this disjoining pressure, particles do not need to contact one another in order to carry an 'effective stress'. This work may lead towards theoretical predictions of the macroscopic load deformation response of montmorillonitic soils based on micromechanical modelling of particles.

Dada uma argila montmorilonítica de alta porosidade e saturada por counteríons monovalentes, investigamos as respostas da argila ao nível de partículas para diferentes cargas externas. Como soluções analíticas não são possíveis para arranjos complexos de partículas, empregamos modelos computacionais micro-mecânicos (baseados na solução das equações de Poisson-Nernst-Planck), utilizando o método de elementos finitos, para estimar counteríons e distribuições de potencial elétrico para partículas em diversos ângulos e distâncias uma da outra. Nós então calculamos as pressões de separação usando a relação de Van't Hoff e a tensão de cisalhamento de Maxwell. À medida que a distância entre as partículas de argila diminui e as duplas camadas se sobrepõem, a concentração de counteríons nos microporos entre as partículas de argila aumenta. Este aumento reduz o potencial químico do fluido nos poros e cria um gradiente de potencial químico no solvente, que gera a chamado pressão 'osmótica' ou de 'separação'. Devido a esta pressão de separação, as partículas não precisam de contato entre si, a fim de exercer uma 'tensão efetiva'. Este trabalho pode conduzir a previsões teóricas da resposta macroscópica a carga de deformação em solos montmoriloníticos baseado na modelação micromecânica das partículas.

Electro-osmosis in kaolinite with pH-dependent surface charge modelling by homogenization

A new three-scale model to describe the coupling between pH-dependent flows and transient ion transport, including adsorption phenomena in kaolinite clays, is proposed. The kaolinite is characterized by three separate nano/micro and macroscopic length scales. The pore (micro)-scale is characterized by micro-pores saturated by an aqueous solution containing four monovalent ions and charged solid particles surrounded by thin electrical double layers. The movement of the ions is governed by the Nernst-Planck equations, and the influence of the double layers upon the flow is dictated by the Helmholtz-Smoluchowski slip boundary condition on the tangential velocity. In addition, an adsorption interface condition for the Na+ transportis postulated to capture its retention in the electrical double layer. Thetwo-scalenano/micro model including salt adsorption and slip boundary condition is homogenized to the Darcy scale and leads to the derivation of macroscopic governing equations. One of the notable features of the three-scale model is there construction of the constitutive law of effective partition coefficient that governs the sodium adsorption in the double layer. To illustrate the feasibility of the three-scale model in simulating soil decontamination by electrokinetics, the macroscopic model is discretized by the finite volume method and the desalination of a kaolinite sample by electrokinetics is simulated.

Neste artigo propomos um modelo em três escalas para descrever o acoplamento entre o fluxo eletroosmótico e o transporte de íons incluindo fenômenos de adsorção em uma caulinita. A argila é caracterizada por três escalas nano/micro e macroscópica. A escala microscópica é constituída por micro-poros saturados por uma solução aquosa contendo quatro íons monovalentes e partículas sólidas carregadas eletricamente circundadas por uma dupla camada elétrica fina. O movimento dos íons é governado pelas equações de Nernst-Planck e a influência da dupla camada sobre o fluxo aquoso é modelada por uma condição de contorno de deslizamento da componente tangencial do campo de velocidade (condição de Helmholtz-Smoluchowski). Além disso, uma condição de adsorção na interface fluido-sólido para os íons Na+ é postulada capturando a retenção do sódio na dupla camada elétrica. O modelo em duas escalas nano/micro incluindo a adsorção do sal e a condição de deslizamento da velocidade é homogeneizado levando a derivação das equações macroscópicas na escala de Darcy. Um dos aspectos inovadores do modelo em três escalas é a reconstrução da lei constitutiva para o coeficiente de partição que governa a adsorção do Na+ na dupla camada elétrica. Para ilustrar as potencialidades do modelo em três escalas na simulação da eletroremediação de solos argilosos, o modelo macroscópico é discretizado utilizando o método de volumes finitos no intuito de simular a dessalinização de uma amostra de caulinita por técnica de eletrocinética.